Showing papers in "Mathematische Zeitschrift in 1999"

The local structure of length spaces with curvature bounded above

Abstract: We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X.

A geometrical approach to monotone functions in R n

Abstract: The paper is concerned with the fine properties of monotone functions on $\mathbb{R}^n$ . We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a $C^{1,\alpha}$ convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set.

Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations

Abstract: We prove a Nekhoroshev type result [26,27] for the nonlinear Schrodinger equation \begin{eqnarray} iu_t=-u_{xx}-mu-u \varphi (|u|^2) , \end{eqnarray} with vanishing or periodic boundary conditions on $[0,\pi]$ ; here $m\in\mathbb{R}$ is a parameter and $\varphi:\mathbb{R}\to\mathbb{R}$ is a function analytic in a neighborhood of the origin and such that $\varphi(0)=0$ , $\varphi'(0) ot=0$ . More precisely, we consider the Cauchy problem for (0.1) with initial data which extend to analytic entire functions of finite order, and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum. The proof is obtained by a method which applies to Hamiltonian perturbations of linear PDE's with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of $\left\Vert u_{L^2}\right\Vert$ , which is an integral of motion. The main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here.

Depth for complexes, and intersection theorems

Abstract: This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules.

The Bergman metric on hyperconvex domains

Abstract: In this article it is shown that the Bergman metric on a bounded hyperconvex domain in $\mathbb{C}^n$ is always complete. A counterexample demonstrates, that the converse conclusion fails in general.

An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres

Abstract: where Ω is a planar domain and f is in the usual Holder space C2,α(Ω). Without loss of generality we shall consider only locally convex solutions of (1). This equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere (in short PA-sphere) in the unimodular affine real 3-space (see [C1], [C2], [CY] and [LSZ]). Contrary to the case of smooth bounded convex domains, little is known about solutions of (1) when the domain is unbounded. Here, we recall a famous result by K. Jorgens which asserts that any solution of (1) on Ω = R2 is a quadratic polynomial (see [J]) and we also mention a previous paper (see [FMM]) where the authors study solutions of (1) on the exterior of a planar domain that are regular at infinity. Since the underlying almost-complex structure of (1) is integrable, one expects PA-spheres (with their canonical conformal structure) to be conveniently described in terms of meromorphic functions. The reader will find in Sect. 2 a complex representation of PA-spheres and, particularly, a complex description for the solutions of (1).

Generic singularities of certain Schubert varieties

Abstract: Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For $w \in W$ , let $X_{wQ}$ denote the Schubert variety $\overline{BwQ}/Q$ . For $y\in W$ such that $X_{yQ}\subseteq X_{wQ}$ , one knows that ByQ / Q admits a T-stable transversal in $X_{wQ}$ , which we denote by ${\cal N}_{yQ,wQ}$ . We prove that, under certain hypotheses, ${\cal N}_{yQ, wQ}$ is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and $X_{yQ}$ is an irreducible component of the boundary of $X_{wQ}$ (that is, the complement of the open orbit of the stabiliser in G of $X_{wQ}$ ). As a consequence, we describe the singularity of $X_{wQ}$ along ByQ / Q and obtain that the boundary of $X_{wQ}$ equals its singular locus.

Solitons and the electromagnetic field

Abstract: In a recent paper [4], it has been introduced a Lorentz invariant equation in three space dimensions, having soliton like solutions. We recall that, roughly speaking, a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations (see [6], [15], [13], [10]). The equation introduced in [4] is the Euler Lagrange equation of an action functional

A finite set of generators for the Kauffman bracket skein algebra

Abstract: If F is a compact orientable surface it is known that the Kauffman bracket skein module of \(F \times I\) has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes \(S^3\).

Inverse problem and the trace formula for the Hill operator, II

Abstract: Consider the Hill operator \(T = -d^2/dx^2+q(x)\) on \(L^2(\mathbb{R}), \) where \(q\in L^2(0,1)\) is a 1-periodic real potential and \(\int _0^1q(x)dx=0.\) The spectrum of T is absolutely continuous and consists of intervals separated by gaps \(\gamma _n=(a^-_n, a^+_n ), n\geq 1\). Let \(\mu_n, n\geq 1,\) be the Dirichlet eigenvalue of the equation \(-y''+qy=ly\) on the interval [0,1]. Introduce the vector \(g_n=(g_{cn}, g_{sn})\in \mathbb{R}^2,\) with components \(g_{cn}=(a_n^++a_n^+)/2-\mu_n,\) and \(g_{sn}=||\gamma _n|^2/4-g_{cn}^2|s_n,\) where the sign \(s_n=+ \) or \(s_n=-\) for all \(n\geq 1\). Using nonlinear functional analysis in Hilbert spaces we show, that the mapping \(g: q\to g(q)=\{g_n \}_1^{\iy }\in \ell^2\oplus \ell^ 2\) is a real analytic isomorphism. In the second part a trace formula for \(q\in L^2(0,1)\) is proved.

A superquadratic indefinite elliptic system and its Morse-Conley-Floer homology.

Abstract: We study critical points of the indefinite functional $f_H(u, v)=\int\{ abla u\cdot abla v- H(x, u, v) \}\mathrm{d} x$ by applying Floer's homology construction to the ordinary gradient flow of the functional f on a suitable Sobolev space. One of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the Floer homology for $f_H$ .

Local conditions insuring bifurcation from the continuous spectrum

Abstract: We consider a family of equations −∆u(x) + λu(x) = f(x, u(x)), λ > 0, x ∈ R , (1)λ where the nonlinearity f : RN ×R → R satisfiesf(x, 0) = 0, a.e.x ∈ RN . We say thatλ = 0 is a bifurcation point for(1)λ if there exists a sequence {(λn, un)} ⊂ R+ × H1(RN ) of nontrivial solutions of(1)λn with λn → 0 and ||un||H1(RN ) → 0. In this case{(λn, un)} is called a bifurcating sequence. The aim of the paper is to show that weak conditions on f(x, .) around zero suffice to guarantee that λ = 0 is a bifurcation point for(1)λ. More precisely suppose there is δ > 0 such that (H1) f : RN × [−δ, δ] → R is Caratheodory. (H2) lim |x|→∞ f(x, s) = 0 uniformly for s ∈ [−δ, δ].

Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth

Abstract: is the homogeneous Sobolev space of order μ. We prove the global solvability of (1.1) in the critical Sobolev spacė Hn/2 with nonlinearityf of exponential type, a typical example of which is given by u2eλ|u| with λ > 0. There is a large literature on the Cauchy problem for the equation (1.1), see for instance [2,5,7,9,10,12,13,16,17,21,22]. It is well-known that the Cauchy problem (1.1) is locally well-posed in the usual Sobolev space Hμ(R) if μ > n/2 andf is any smooth function withf(0) = 0 [16], or if 1/2 ≤ μ < n/2 andf is given as a single power nonlinearity λ|u|p−1uwith λ ∈ Candp ≤ 1+4/(n−2μ) [9,13,17,22]. Moreover if p = 1+4/(n−2μ) and1/2 ≤ μ < n/2, then we have global̇ Hμ-solutions with the Cauchy

On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture

Abstract: In [Arn, Appendix 9] Arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain (i.e., exact or Hamiltonian) selfdiffeomorphisms of a closed symplectic manifold (M,ω) and the minimum number of critical points of any smooth (= C∞) function on M . The first author succeeded in proving this form of the Arnold conjecture [R2] under the hypothesis that ω and c1 vanish on all spherical homology classes and that there is equality between the Lusternik–Schnirelmann category of M and the dimension of M . In this paper, we use a fundamental property of category weight to show that, for any closed symplectic manifold whose symplectic form vanishes on the image of the Hurewicz map, the required equality holds. Thus, we show that the original form of the Arnold Conjecture holds for all symplectic manifolds having ω|π2(M) = 0 = c1|π2(M).

On estimates in Hardy spaces for the Stokes flow in a half space

Abstract: Since $u=e^{-tA}u_{0}$ and $||A^{1/2}u||_{2}=||\ abla u||_{2},$ (2) follows for $p=2.(\\mathrm{S}\\mathrm{e}\\mathrm{e}$ Borchers and Miyakawa [3] for applications.) For $1

Generation of k-jets on toric varieties

Abstract: The notion of a k-convex \(\Delta\)-support function for a toric variety \(X(\Delta)\) is introduced A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the \(\Delta\)-support function \(\psi_L\) Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k

Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds

Abstract: We consider the Dirac operator on compact quaternionic Kahler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

A duality property for complex Lie algebroids

Abstract: Interpreting Lie algebroid theory in terms of ${\cal D}$ -modules, we define a duality functor for a Lie algebroid as well as a direct image functor for a morphism of Lie algebroids. Generalizing the work of Schneiders (see also the work of Schapira-Schneiders) and making assumptions analog to his, we show that the duality functor and the direct image functor commute. As an application, we extend to Lie algebroids some duality properties already known for Lie algebras.

The Kelvin–Nevanlinna–Royden criterion for p -parabolicity

Abstract: We generalize the so called Kelvin–Nevanlinna–Royden criterion for the parabolicity of manifolds to the case of p-parabolicity for all \(1 < p < \infty\).

On analytical applications of stable homotopy (the Arnold conjecture, critical points)

Abstract: We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and catM = dimM Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity” property

Centers of reduced enveloping algebras

Abstract: We compute the inverse image of a functional in the Zassenhaus variety. We apply this computation to describe the category of representations for a regular functional.

The weak conjecture on spherical classes

Abstract: Let \(\mathcal{A}\) be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, \(\) which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, \(\Gamma_k^{\wedge}\), to \({\bf F}_2[x_1^{\pm 1},\ldots ,x_k^{\pm 1}]\) and is the inclusion of the Dickson algebra, \(D_k \subset \Gamma_k^{\wedge}\), into \({\bf F}_2[x_1,\ldots ,x_k]\). This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in \(Tor_k^{{\mathcal{A}}}({\bf F}_2,{\bf F}_2)\)}, (2) or a conjecture on ${\mathcal{A}}$ -decomposability of the Dickson algebra in $\Gamma_k^{\wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2).

Boundaries of geometrically finite groups

Abstract: We show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected. On the other hand, if there is a separating horoball centred on a parabolic point, one obtains a non-trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups. Together with results from elsewhere, one deduces that if $ \Gamma $ is a relatively hyperbolic group such that each maximal parabolic subgroup is one-or-two ended, finitely presented, and contains no infinite torsion subgroup, then the boundary of $ \Gamma $ is locally connected if it is connected. As a corollary, we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected.

Generalized Jacobians of spectral curves and completely integrable systems

Abstract: Consider an ordinary differential equation which has a Lax pair representation \(\dot{A}(x)= [A(x),B(x)]\), where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold \(\) of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve \(\{(x,y): P(x,y)=0 \}\) with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant.

Representation type of Schur algebras

Abstract: We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics.

Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds

Abstract: In this paper we study a nonlinear elliptic system of equations imposed on a map from a complete Hermitian (non-Kahler) manifold to a Riemannian manifold. This system is more appropriate to Hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are Hermitian harmonic maps. It was first studied by Jost and Yau in [J-Y], and was applied to study the rigidity of compact Hermitian manifolds. We extend their existence and uniqueness results to the case where both domain and target manifolds are complete. Hopefully the results will be useful to study corresponding rigidity of complete Hermitian manifolds. Let M be a complex manifold with Hermitian metric (hαβ), and let N be a Riemannian manifold with metric (gij) and Christoffel symbols Γ i jk. A Hermitian harmonic map u : M → N satisfies the following elliptic system